MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: average(x,y) -> if(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) if(false(),b1,b2,b3,x,y) -> average(p(x),s(y)) if(true(),b1,b2,b3,x,y) -> if2(b1,b2,b3,x,y) if2(false(),b2,b3,x,y) -> if3(b2,b3,x,y) if2(true(),b2,b3,x,y) -> 0() if3(false(),b3,x,y) -> if4(b3,x,y) if3(true(),b3,x,y) -> 0() if4(false(),x,y) -> average(s(x),p(p(y))) if4(true(),x,y) -> s(0()) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {average,if,if2,if3,if4,le,p} and constructors {0,false,s ,true} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() - Strict TRS: average(x,y) -> if(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) if(false(),b1,b2,b3,x,y) -> average(p(x),s(y)) if(true(),b1,b2,b3,x,y) -> if2(b1,b2,b3,x,y) if2(false(),b2,b3,x,y) -> if3(b2,b3,x,y) if2(true(),b2,b3,x,y) -> 0() if3(false(),b3,x,y) -> if4(b3,x,y) if3(true(),b3,x,y) -> 0() if4(false(),x,y) -> average(s(x),p(p(y))) if4(true(),x,y) -> s(0()) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() * Step 3: WeightGap MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() - Strict TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(p) = {1}, uargs(average#) = {1,2}, uargs(if#) = {1,2,3,4}, uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(average) = [0] p(false) = [0] p(if) = [0] p(if2) = [0] p(if3) = [0] p(if4) = [0] p(le) = [2] x2 + [1] p(p) = [1] x1 + [1] p(s) = [1] x1 + [1] p(true) = [0] p(average#) = [1] x1 + [1] x2 + [0] p(if#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [3] p(if2#) = [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(if3#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [7] p(if4#) = [1] x2 + [1] x3 + [0] p(le#) = [2] x1 + [7] x2 + [2] p(p#) = [2] x1 + [4] p(c_1) = [1] x1 + [1] p(c_2) = [1] x1 + [1] p(c_3) = [1] x1 + [7] p(c_4) = [1] x1 + [2] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [1] x1 + [1] p(c_9) = [0] p(c_10) = [4] p(c_11) = [1] p(c_12) = [1] x1 + [5] p(c_13) = [0] p(c_14) = [0] Following rules are strictly oriented: if3#(false(),b3,x,y) = [1] b3 + [1] x + [1] y + [7] > [1] x + [1] y + [0] = c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) = [1] b3 + [1] x + [1] y + [7] > [1] = c_7() le#(s(x),0()) = [2] x + [4] > [1] = c_11() le#(s(x),s(y)) = [2] x + [7] y + [11] > [2] x + [7] y + [7] = c_12(le#(x,y)) p#(0()) = [4] > [0] = c_13() p#(s(x)) = [2] x + [6] > [0] = c_14() le(0(),y) = [2] y + [1] > [0] = true() le(s(x),0()) = [1] > [0] = false() le(s(x),s(y)) = [2] y + [3] > [2] y + [1] = le(x,y) p(0()) = [1] > [0] = 0() p(s(x)) = [1] x + [2] > [1] x + [0] = x Following rules are (at-least) weakly oriented: average#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [14] = c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) = [1] b1 + [1] b2 + [1] b3 + [1] x + [1] y + [3] >= [1] x + [1] y + [3] = c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) = [1] b1 + [1] b2 + [1] b3 + [1] x + [1] y + [3] >= [1] b2 + [1] b3 + [1] x + [1] y + [7] = c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) = [1] b2 + [1] b3 + [1] x + [1] y + [0] >= [1] b2 + [1] b3 + [1] x + [1] y + [9] = c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) = [1] b2 + [1] b3 + [1] x + [1] y + [0] >= [1] = c_5() if4#(false(),x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [4] = c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) = [1] x + [1] y + [0] >= [0] = c_9() le#(0(),y) = [7] y + [2] >= [4] = c_10() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() - Weak DPs: if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4,5} by application of Pre({4,5}) = {3}. Here rules are labelled as follows: 1: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) 2: if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) 3: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) 4: if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) 5: if2#(true(),b2,b3,x,y) -> c_5() 6: if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) 7: if4#(true(),x,y) -> c_9() 8: le#(0(),y) -> c_10() 9: if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) 10: if3#(true(),b3,x,y) -> c_7() 11: le#(s(x),0()) -> c_11() 12: le#(s(x),s(y)) -> c_12(le#(x,y)) 13: p#(0()) -> c_13() 14: p#(s(x)) -> c_14() * Step 5: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() - Weak DPs: if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3} by application of Pre({3}) = {1}. Here rules are labelled as follows: 1: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) 2: if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) 3: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) 4: if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) 5: if4#(true(),x,y) -> c_9() 6: le#(0(),y) -> c_10() 7: if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) 8: if2#(true(),b2,b3,x,y) -> c_5() 9: if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) 10: if3#(true(),b3,x,y) -> c_7() 11: le#(s(x),0()) -> c_11() 12: le#(s(x),s(y)) -> c_12(le#(x,y)) 13: p#(0()) -> c_13() 14: p#(s(x)) -> c_14() * Step 6: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() - Weak DPs: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) -->_1 if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)):6 -->_1 if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))):2 2:S:if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)):1 3:S:if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)):1 4:S:if4#(true(),x,y) -> c_9() 5:S:le#(0(),y) -> c_10() 6:W:if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) -->_1 if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)):7 -->_1 if2#(true(),b2,b3,x,y) -> c_5():8 7:W:if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) -->_1 if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)):9 -->_1 if3#(true(),b3,x,y) -> c_7():10 8:W:if2#(true(),b2,b3,x,y) -> c_5() 9:W:if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) -->_1 if4#(true(),x,y) -> c_9():4 -->_1 if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))):3 10:W:if3#(true(),b3,x,y) -> c_7() 11:W:le#(s(x),0()) -> c_11() 12:W:le#(s(x),s(y)) -> c_12(le#(x,y)) -->_1 le#(s(x),s(y)) -> c_12(le#(x,y)):12 -->_1 le#(s(x),0()) -> c_11():11 -->_1 le#(0(),y) -> c_10():5 13:W:p#(0()) -> c_13() 14:W:p#(s(x)) -> c_14() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: p#(s(x)) -> c_14() 13: p#(0()) -> c_13() 11: le#(s(x),0()) -> c_11() 8: if2#(true(),b2,b3,x,y) -> c_5() 10: if3#(true(),b3,x,y) -> c_7() * Step 7: Decompose MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() - Weak DPs: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) le#(s(x),s(y)) -> c_12(le#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() - Weak DPs: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) le#(0(),y) -> c_10() le#(s(x),s(y)) -> c_12(le#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0 ,false/0,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0 ,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} Problem (S) - Strict DPs: le#(0(),y) -> c_10() - Weak DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() le#(s(x),s(y)) -> c_12(le#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0 ,false/0,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0 ,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} ** Step 7.a:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() - Weak DPs: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) le#(0(),y) -> c_10() le#(s(x),s(y)) -> c_12(le#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) -->_1 if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)):6 -->_1 if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))):2 2:S:if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)):1 3:S:if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)):1 4:S:if4#(true(),x,y) -> c_9() 5:W:le#(0(),y) -> c_10() 6:W:if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) -->_1 if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)):7 7:W:if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) -->_1 if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)):9 9:W:if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) -->_1 if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))):3 -->_1 if4#(true(),x,y) -> c_9():4 12:W:le#(s(x),s(y)) -> c_12(le#(x,y)) -->_1 le#(0(),y) -> c_10():5 -->_1 le#(s(x),s(y)) -> c_12(le#(x,y)):12 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 12: le#(s(x),s(y)) -> c_12(le#(x,y)) 5: le#(0(),y) -> c_10() ** Step 7.a:2: PredecessorEstimationCP MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() - Weak DPs: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 4: if4#(true(),x,y) -> c_9() The strictly oriented rules are moved into the weak component. *** Step 7.a:2.a:1: NaturalMI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() - Weak DPs: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {average#,if#,if2#,if3#,if4#,le#,p#} TcT has computed the following interpretation: p(0) = [2] p(average) = [8] x1 + [1] x2 + [2] p(false) = [0] p(if) = [2] x1 + [1] x2 + [1] x3 + [2] x5 + [2] p(if2) = [1] x3 + [1] x4 + [1] x5 + [1] p(if3) = [1] x2 + [1] x3 + [1] x4 + [0] p(if4) = [1] x1 + [2] x2 + [0] p(le) = [8] x2 + [0] p(p) = [0] p(s) = [0] p(true) = [0] p(average#) = [1] p(if#) = [1] p(if2#) = [1] p(if3#) = [1] p(if4#) = [1] p(le#) = [2] x1 + [1] p(p#) = [4] x1 + [8] p(c_1) = [1] x1 + [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [2] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [4] p(c_11) = [2] p(c_12) = [1] x1 + [1] p(c_13) = [1] p(c_14) = [1] Following rules are strictly oriented: if4#(true(),x,y) = [1] > [0] = c_9() Following rules are (at-least) weakly oriented: average#(x,y) = [1] >= [1] = c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) = [1] >= [1] = c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) = [1] >= [1] = c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) = [1] >= [1] = c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) = [1] >= [1] = c_6(if4#(b3,x,y)) if4#(false(),x,y) = [1] >= [1] = c_8(average#(s(x),p(p(y)))) *** Step 7.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) - Weak DPs: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if4#(true(),x,y) -> c_9() - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 7.a:2.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) - Weak DPs: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if4#(true(),x,y) -> c_9() - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) -->_1 if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)):4 -->_1 if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))):2 2:S:if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)):1 3:S:if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)):1 4:W:if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) -->_1 if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)):5 5:W:if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) -->_1 if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)):6 6:W:if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) -->_1 if4#(true(),x,y) -> c_9():7 -->_1 if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))):3 7:W:if4#(true(),x,y) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: if4#(true(),x,y) -> c_9() *** Step 7.a:2.b:2: Failure MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) - Weak DPs: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(0(),y) -> c_10() - Weak DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) if4#(true(),x,y) -> c_9() le#(s(x),s(y)) -> c_12(le#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:le#(0(),y) -> c_10() 2:W:average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) -->_1 if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)):4 -->_1 if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))):3 3:W:if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)):2 4:W:if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) -->_1 if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)):5 5:W:if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) -->_1 if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)):6 6:W:if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) -->_1 if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))):7 -->_1 if4#(true(),x,y) -> c_9():8 7:W:if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)):2 8:W:if4#(true(),x,y) -> c_9() 9:W:le#(s(x),s(y)) -> c_12(le#(x,y)) -->_1 le#(s(x),s(y)) -> c_12(le#(x,y)):9 -->_1 le#(0(),y) -> c_10():1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) 7: if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) 6: if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) 5: if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) 4: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) 3: if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) 8: if4#(true(),x,y) -> c_9() ** Step 7.b:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(0(),y) -> c_10() - Weak DPs: le#(s(x),s(y)) -> c_12(le#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: le#(0(),y) -> c_10() le#(s(x),s(y)) -> c_12(le#(x,y)) ** Step 7.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(0(),y) -> c_10() - Weak DPs: le#(s(x),s(y)) -> c_12(le#(x,y)) - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: le#(0(),y) -> c_10() The strictly oriented rules are moved into the weak component. *** Step 7.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(0(),y) -> c_10() - Weak DPs: le#(s(x),s(y)) -> c_12(le#(x,y)) - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_12) = {1} Following symbols are considered usable: {average#,if#,if2#,if3#,if4#,le#,p#} TcT has computed the following interpretation: p(0) = [1] p(average) = [0] p(false) = [0] p(if) = [0] p(if2) = [0] p(if3) = [0] p(if4) = [0] p(le) = [0] p(p) = [0] p(s) = [1] x1 + [2] p(true) = [0] p(average#) = [0] p(if#) = [0] p(if2#) = [0] p(if3#) = [0] p(if4#) = [0] p(le#) = [8] x1 + [1] x2 + [3] p(p#) = [0] p(c_1) = [2] p(c_2) = [1] x1 + [1] p(c_3) = [1] p(c_4) = [1] x1 + [4] p(c_5) = [4] p(c_6) = [4] x1 + [0] p(c_7) = [2] p(c_8) = [0] p(c_9) = [8] p(c_10) = [3] p(c_11) = [8] p(c_12) = [1] x1 + [4] p(c_13) = [1] p(c_14) = [2] Following rules are strictly oriented: le#(0(),y) = [1] y + [11] > [3] = c_10() Following rules are (at-least) weakly oriented: le#(s(x),s(y)) = [8] x + [1] y + [21] >= [8] x + [1] y + [7] = c_12(le#(x,y)) *** Step 7.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: le#(0(),y) -> c_10() le#(s(x),s(y)) -> c_12(le#(x,y)) - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 7.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: le#(0(),y) -> c_10() le#(s(x),s(y)) -> c_12(le#(x,y)) - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:le#(0(),y) -> c_10() 2:W:le#(s(x),s(y)) -> c_12(le#(x,y)) -->_1 le#(s(x),s(y)) -> c_12(le#(x,y)):2 -->_1 le#(0(),y) -> c_10():1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: le#(s(x),s(y)) -> c_12(le#(x,y)) 1: le#(0(),y) -> c_10() *** Step 7.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). MAYBE